Discrete Chaos

Rating:
Summary: A notable entry with a new perspective
Review: "Discrete Chaos" resembles in many ways the highly successful 1989 introductory text by Robert Devaney on chaotic dyanamical systems. However, a great deal has happened since Devaney's text appeared, and Discrete Chaos includes some of the new finds (e.g., the new results on stability of maps on the real line and the author's own work on the converse of Sharkovsky's Theorem). Discrete Chaos also adds a new element unique to it: The author's perspective as a successful researcher and a talented expositor in the area of difference equations. This is important since it balances the attention between the familiar topological/algebraic ideas and the more purely analytical results. Equally important perhaps, is the author's talent for writing introductory level books (he also wrote the very readable "An Introduction to Difference Equations"). Discrete Chaos contains many interesting examples and helpful exercises, although the proofs of the more technical results are not given.

Rating:
Summary: A notable entry with a new perspective
Review: "Discrete Chaos" resembles in many ways the highly successful 1989 introductory text by Robert Devaney on chaotic dyanamical systems. However, a great deal has happened since Devaney's text appeared, and Discrete Chaos includes some of the new finds (e.g., the new results on stability of maps on the real line and the author's own work on the converse of Sharkovsky's Theorem). Discrete Chaos also adds a new element unique to it: The author's perspective as a successful researcher and a talented expositor in the area of difference equations. This is important since it balances the attention between the familiar topological/algebraic ideas and the more purely analytical results. Equally important perhaps, is the author's talent for writing introductory level books (he also wrote the very readable "An Introduction to Difference Equations"). Discrete Chaos contains many interesting examples and helpful exercises, although the proofs of the more technical results are not given.

Rating:
Summary: Accessible Chaos Theory
Review: Discrete chaos is an excellent text. It can serve both as an introductory text in discrete-time dynamical systems (chaos theory) and as a resource for more advanced work. The book provides an accessible introduction to discrete-time dynamical systems with many interesting applications.

The book is divided into six sections. Section 1 and Section 2 introduce essential concepts for describing and analyzing chaotic discrete-time systems. Section 3 and Section 5 focus on chaos in one dimension and two dimensions, respectively. Section 4 describes the stability of two-dimensional systems. Fractals are formulated and analyzed in Section 6. Section 7 is devoted to the study of The Julia and Mandelbrot Sets.

All chapters end with excellent exercises. The book also has answers to the exercises. The book is designed to be independent of any particular computing environment.

Rating:
Summary: Truely Exceptional: Order within chaos
Review: This book is highly recommended to all those who aspire to enjoy the world of chaos and fractals. It is written by one of the masters of discrete dynamics and difference equations. I was sold on Devaney's book until I saw this book. Moreover, I am not fully satisfied with the explanations and the writing style in the book by Alligood, Sauer and Yorke. At last we have some genuine and realistic applications to discrete dynamical systems. The best bargain of all is the chapter on Iterated function systems; the book provides a delightful and very readable , yet mathematically honest, account. Almost all the results in the book are proved completely and clearly. I enjoyed the relaxed and lucid writing style of this author. Keep them coming and way to go Elaydi!